1. Field of the Invention
The present invention relates to multi-point model reductions of VSLI interconnects, and more particularly to multi-point model reductions of VLSI interconnects using rational arnoldi method with adaptive orders (RAMAO).
2. Description of Related Art
Interconnect plays a significant role in the recent development of high-speed VLSI design. Due to the continuous increasing in component densities and clock rates, the signal integrity problems naturally arise in the interconnect structure. For efficient simulations, it is necessary to construct a low-order macro-model whose terminal behaviors essentially capture the complicated interactions among the interconnects. The process of finding such a reduced network is referred to as model reduction [References 4, 6, 13-15 and 18].
Recently, several methods that are based on Pade synthesis have been applied to improve the model-order reduction techniques. Asymptotic waveform evaluation (AWE) [References 3 and 15], Pade via Lanczos (PVL) [Reference 6 and 7], and congruence transformations (CT) [Reference 13] have successfully been used to analyze interconnect systems. Among all existing methods, it has been shown that the class of Krylov-space methods seems to be more accurate, because it can avoid the ill-conditional problems. However, these conventional approximation methods tend to converge in a local fashion around a single frequency, because Pade approximation is exact at the point while accuracy is lost away from it. So the reduced-order model may grow large before becoming an acceptable global approximation. To overcome this difficulty, multi-point Pade has been proposed [References 2, 5, 12 and 18].
The straightforward way for multi-point moment matching applications is to apply the Krylov subspace algorithm at various expansion frequencies. This is the so-called rational Krylov algorithm [references 1, 9 and 16]. The present invention uses the RAMAO to simplify the conventional algorithm without determining the order of moments at each expansion frequency in advance. The concept was first developed in the rational Lanczos method [reference 8]. In this work, the exact error between the output moment of the original system and that of the reduced-order system, associated with each expansion point, can be determined explicitly. In each iteration of the proposed method, the expansion frequency corresponding to the maximum output moment error is chosen. Consequently, the corresponding reduced-order model will yield the greatest improvement in output moments among all reduced-order models of the same order.